AT-OEMO-FED1 2020 Problem 3

On a blackboard there are three positive integers. In each step the three numbers on the board are denoted as $a, b, c$ such that $a >gcd(b, c)$ , then $a$ gets replaced by $a-gcd(b, c)$ . The game ends if there is no way to denote the numbers such that $a >gcd(b, c)$ .

Prove that the game always ends and that the last three numbers on the blackboard only depend on the starting numbers.

(Theresia Eisenkölbl)